Problem 75 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ \left|-\frac{2}{3}\right| $$

Step-by-Step Solution

Answer

\(\frac{2}{3}\)

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means the absolute value of any real number is always non-negative.

Step 2: Evaluate the Absolute Value

For the expression \(-\frac{2}{3}\), we are asked to find its absolute value. This involves removing any negative sign, as absolute values are always positive or zero. \(\left| -\frac{2}{3} \right| = \frac{2}{3}\).

Key Concepts

Number Line

A number line is a straight line that visually represents numbers. It extends infinitely in both directions, where numbers increase from left to right. Zero is located at the center, with positive numbers to the right and negative numbers to the left. The number line provides a simple way to visualize arithmetic operations and concepts such as absolute value.

Using the number line, absolute value can be understood as the distance of a number from zero. Since distance is always non-negative, absolute value is never negative. For example, both \(-3\) and \(3\) are three units away from zero, so their absolute value is \(3\).

This representation helps in understanding how the absolute value functions work, acting as distances rather than directions or signs. So, whether a number is \(-\frac{2}{3}\) or \(\frac{2}{3}\), its absolute value remains \(\frac{2}{3}\).

Non-Negative

In mathematics, a non-negative number is a number that is either positive or zero. The absolute value of numbers gives us their non-negative form. It strips away any negative signs, indicating only the magnitude or size of the number without any directional component.

Understanding that absolute values are always non-negative simplifies tasks where only the size matters, not the direction. For example, when finding the absolute value of \(-\frac{2}{3}\), we ignore its negative sign and see it as simply \(\frac{2}{3}\). This concept is crucial in real-world applications like measuring distances, where negative values are non-sensible.

• Absolute values remove the negative sign.
• They transform negative numbers to their positive form.
• Zer0 remains zero since it's non-negative.

The concept of non-negativity is fundamental in absolute value operations since it ensures results that are usable and meaningful in practical situations.

Real Number

Real numbers comprise both rational and irrational numbers, representing the entirety of the number line. This includes any number you can think of that lies on the number line, whether whole, fractional, or irrational.

Rational numbers are those that can be expressed as fractions, such as \(\frac{2}{3}\), and include integers and whole numbers. Irrational numbers cannot be expressed as fractions, they continue indefinitely without repeating, like \(\pi\) or \(\sqrt{2}\).

Real numbers are crucial when considering absolute values. The reason is that absolute values apply to any real number, essentially treating them all as distances from zero on the number line. Whether a number is positive or negative, a whole number, or a fraction like \(-\frac{2}{3}\), we can always express its absolute value since real numbers cover everything on the line.

• Real numbers are comprehensive and include:
  • Rational numbers (e.g., integers, fractions)
  • Irrational numbers (e.g., \(\pi\), \(\sqrt{2}\))
• Absolute values work with any real number to ensure it represents a non-negative distance.

This makes real numbers and their absolute values extensively applicable in mathematics and beyond.